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Theorem rankxplim3 8744

Description: The rank of a Cartesian product is a limit ordinal iff its union is. (Contributed by NM, 19-Sep-2006.)

**Hypotheses**
Ref	Expression
rankxplim.1	⊢ 𝐴 ∈ V
rankxplim.2	⊢ 𝐵 ∈ V

**Assertion**
Ref	Expression
rankxplim3	⊢ (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim ∪ (rank‘(𝐴 × 𝐵)))

Proof of Theorem rankxplim3 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limuni2 5786	. 2 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim ∪ (rank‘(𝐴 × 𝐵)))
2		0ellim 5787	. . . 4 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ∅ ∈ ∪ (rank‘(𝐴 × 𝐵)))
3		n0i 3920	. . . 4 ⊢ (∅ ∈ ∪ (rank‘(𝐴 × 𝐵)) → ¬ ∪ (rank‘(𝐴 × 𝐵)) = ∅)
4		unieq 4444	. . . . . 6 ⊢ ((rank‘(𝐴 × 𝐵)) = ∅ → ∪ (rank‘(𝐴 × 𝐵)) = ∪ ∅)
5		uni0 4465	. . . . . 6 ⊢ ∪ ∅ = ∅
6	4, 5	syl6eq 2672	. . . . 5 ⊢ ((rank‘(𝐴 × 𝐵)) = ∅ → ∪ (rank‘(𝐴 × 𝐵)) = ∅)
7	6	con3i 150	. . . 4 ⊢ (¬ ∪ (rank‘(𝐴 × 𝐵)) = ∅ → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
8	2, 3, 7	3syl 18	. . 3 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
9		rankon 8658	. . . . . . . . . 10 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ On
10	9	onsuci 7038	. . . . . . . . 9 ⊢ suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
11	10	onsuci 7038	. . . . . . . 8 ⊢ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
12	11	elexi 3213	. . . . . . 7 ⊢ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ V
13	12	sucid 5804	. . . . . 6 ⊢ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ suc suc suc (rank‘(𝐴 ∪ 𝐵))
14	11	onsuci 7038	. . . . . . . 8 ⊢ suc suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
15		ontri1 5757	. . . . . . . 8 ⊢ ((suc suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On ∧ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On) → (suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) ↔ ¬ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ suc suc suc (rank‘(𝐴 ∪ 𝐵))))
16	14, 11, 15	mp2an 708	. . . . . . 7 ⊢ (suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) ↔ ¬ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ suc suc suc (rank‘(𝐴 ∪ 𝐵)))
17	16	con2bii 347	. . . . . 6 ⊢ (suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ suc suc suc (rank‘(𝐴 ∪ 𝐵)) ↔ ¬ suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)))
18	13, 17	mpbi 220	. . . . 5 ⊢ ¬ suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵))
19		rankxplim.1	. . . . . . 7 ⊢ 𝐴 ∈ V
20		rankxplim.2	. . . . . . 7 ⊢ 𝐵 ∈ V
21	19, 20	rankxpu 8739	. . . . . 6 ⊢ (rank‘(𝐴 × 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵))
22		sstr 3611	. . . . . 6 ⊢ ((suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)) ∧ (rank‘(𝐴 × 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵))) → suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)))
23	21, 22	mpan2 707	. . . . 5 ⊢ (suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)) → suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)))
24	18, 23	mto 188	. . . 4 ⊢ ¬ suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵))
25		reeanv 3107	. . . . 5 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ On ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦) ↔ (∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦))
26		simprl 794	. . . . . . . . . . . . 13 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥)
27		simpr 477	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥) → (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥)
28		rankuni 8726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ (rank‘∪ (𝐴 × 𝐵))
29		rankuni 8726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (rank‘∪ (𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵))
30	29	unieqi 4445	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ∪ (rank‘∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
31	28, 30	eqtri 2644	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
32		df-ne 2795	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (𝐴 × 𝐵) = ∅)
33	19, 20	xpex 6962	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐴 × 𝐵) ∈ V
34	33	rankeq0 8724	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
35	34	notbii 310	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ (𝐴 × 𝐵) = ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
36	32, 35	bitr2i 265	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ ↔ (𝐴 × 𝐵) ≠ ∅)
37	8, 36	sylib 208	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
38		unixp 5668	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))
39	37, 38	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))
40	39	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (rank‘∪ ∪ (𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
41	31, 40	syl5reqr 2671	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (rank‘(𝐴 ∪ 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)))
42		eqimss 3657	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)) → (rank‘(𝐴 ∪ 𝐵)) ⊆ ∪ ∪ (rank‘(𝐴 × 𝐵)))
43	41, 42	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (rank‘(𝐴 ∪ 𝐵)) ⊆ ∪ ∪ (rank‘(𝐴 × 𝐵)))
44	43	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥) → (rank‘(𝐴 ∪ 𝐵)) ⊆ ∪ ∪ (rank‘(𝐴 × 𝐵)))
45	27, 44	eqsstr3d 3640	. . . . . . . . . . . . . . . . 17 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥) → suc 𝑥 ⊆ ∪ ∪ (rank‘(𝐴 × 𝐵)))
46	45	adantrr 753	. . . . . . . . . . . . . . . 16 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc 𝑥 ⊆ ∪ ∪ (rank‘(𝐴 × 𝐵)))
47		limuni 5785	. . . . . . . . . . . . . . . . 17 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ∪ (rank‘(𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)))
48	47	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → ∪ (rank‘(𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)))
49	46, 48	sseqtr4d 3642	. . . . . . . . . . . . . . 15 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc 𝑥 ⊆ ∪ (rank‘(𝐴 × 𝐵)))
50		vex 3203	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
51		rankon 8658	. . . . . . . . . . . . . . . . . 18 ⊢ (rank‘(𝐴 × 𝐵)) ∈ On
52	51	onordi 5832	. . . . . . . . . . . . . . . . 17 ⊢ Ord (rank‘(𝐴 × 𝐵))
53		orduni 6994	. . . . . . . . . . . . . . . . 17 ⊢ (Ord (rank‘(𝐴 × 𝐵)) → Ord ∪ (rank‘(𝐴 × 𝐵)))
54	52, 53	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ Ord ∪ (rank‘(𝐴 × 𝐵))
55		ordelsuc 7020	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ V ∧ Ord ∪ (rank‘(𝐴 × 𝐵))) → (𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc 𝑥 ⊆ ∪ (rank‘(𝐴 × 𝐵))))
56	50, 54, 55	mp2an 708	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc 𝑥 ⊆ ∪ (rank‘(𝐴 × 𝐵)))
57	49, 56	sylibr 224	. . . . . . . . . . . . . 14 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → 𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)))
58		limsuc 7049	. . . . . . . . . . . . . . 15 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc 𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵))))
59	58	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → (𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc 𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵))))
60	57, 59	mpbid 222	. . . . . . . . . . . . 13 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc 𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)))
61	26, 60	eqeltrd 2701	. . . . . . . . . . . 12 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → (rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵)))
62		limsuc 7049	. . . . . . . . . . . . 13 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ((rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc (rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵))))
63	62	adantr 481	. . . . . . . . . . . 12 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → ((rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc (rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵))))
64	61, 63	mpbid 222	. . . . . . . . . . 11 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc (rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵)))
65		ordsucelsuc 7022	. . . . . . . . . . . 12 ⊢ (Ord ∪ (rank‘(𝐴 × 𝐵)) → (suc (rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ suc ∪ (rank‘(𝐴 × 𝐵))))
66	54, 65	ax-mp 5	. . . . . . . . . . 11 ⊢ (suc (rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ suc ∪ (rank‘(𝐴 × 𝐵)))
67	64, 66	sylib 208	. . . . . . . . . 10 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ suc ∪ (rank‘(𝐴 × 𝐵)))
68		onsucuni2 7034	. . . . . . . . . . . 12 ⊢ (((rank‘(𝐴 × 𝐵)) ∈ On ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦) → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
69	51, 68	mpan 706	. . . . . . . . . . 11 ⊢ ((rank‘(𝐴 × 𝐵)) = suc 𝑦 → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
70	69	ad2antll 765	. . . . . . . . . 10 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
71	67, 70	eleqtrd 2703	. . . . . . . . 9 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐴 × 𝐵)))
72	11, 51	onsucssi 7041	. . . . . . . . 9 ⊢ (suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐴 × 𝐵)) ↔ suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
73	71, 72	sylib 208	. . . . . . . 8 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
74	73	ex 450	. . . . . . 7 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦) → suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵))))
75	74	a1d 25	. . . . . 6 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦) → suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))))
76	75	rexlimdvv 3037	. . . . 5 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (∃𝑥 ∈ On ∃𝑦 ∈ On ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦) → suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵))))
77	25, 76	syl5bir 233	. . . 4 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ((∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦) → suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵))))
78	24, 77	mtoi 190	. . 3 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ¬ (∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦))
79		ianor 509	. . . . . 6 ⊢ (¬ (∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦) ↔ (¬ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∨ ¬ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦))
80		un00 4011	. . . . . . . . . . . . . 14 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 ∪ 𝐵) = ∅)
81		olc 399	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = ∅ → (𝐴 = ∅ ∨ 𝐵 = ∅))
82	81	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅))
83	80, 82	sylbir 225	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 = ∅ ∨ 𝐵 = ∅))
84		xpeq0 5554	. . . . . . . . . . . . 13 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅))
85	83, 84	sylibr 224	. . . . . . . . . . . 12 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
86	85	con3i 150	. . . . . . . . . . 11 ⊢ (¬ (𝐴 × 𝐵) = ∅ → ¬ (𝐴 ∪ 𝐵) = ∅)
87	35, 86	sylbir 225	. . . . . . . . . 10 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → ¬ (𝐴 ∪ 𝐵) = ∅)
88	19, 20	unex 6956	. . . . . . . . . . . 12 ⊢ (𝐴 ∪ 𝐵) ∈ V
89	88	rankeq0 8724	. . . . . . . . . . 11 ⊢ ((𝐴 ∪ 𝐵) = ∅ ↔ (rank‘(𝐴 ∪ 𝐵)) = ∅)
90	89	notbii 310	. . . . . . . . . 10 ⊢ (¬ (𝐴 ∪ 𝐵) = ∅ ↔ ¬ (rank‘(𝐴 ∪ 𝐵)) = ∅)
91	87, 90	sylib 208	. . . . . . . . 9 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → ¬ (rank‘(𝐴 ∪ 𝐵)) = ∅)
92	9	onordi 5832	. . . . . . . . . . 11 ⊢ Ord (rank‘(𝐴 ∪ 𝐵))
93		ordzsl 7045	. . . . . . . . . . 11 ⊢ (Ord (rank‘(𝐴 ∪ 𝐵)) ↔ ((rank‘(𝐴 ∪ 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 ∪ 𝐵))))
94	92, 93	mpbi 220	. . . . . . . . . 10 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 ∪ 𝐵)))
95	94	3ori 1388	. . . . . . . . 9 ⊢ ((¬ (rank‘(𝐴 ∪ 𝐵)) = ∅ ∧ ¬ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥) → Lim (rank‘(𝐴 ∪ 𝐵)))
96	91, 95	sylan 488	. . . . . . . 8 ⊢ ((¬ (rank‘(𝐴 × 𝐵)) = ∅ ∧ ¬ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥) → Lim (rank‘(𝐴 ∪ 𝐵)))
97	96	ex 450	. . . . . . 7 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (¬ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 → Lim (rank‘(𝐴 ∪ 𝐵))))
98		ordzsl 7045	. . . . . . . . . 10 ⊢ (Ord (rank‘(𝐴 × 𝐵)) ↔ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦 ∨ Lim (rank‘(𝐴 × 𝐵))))
99	52, 98	mpbi 220	. . . . . . . . 9 ⊢ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦 ∨ Lim (rank‘(𝐴 × 𝐵)))
100	99	3ori 1388	. . . . . . . 8 ⊢ ((¬ (rank‘(𝐴 × 𝐵)) = ∅ ∧ ¬ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦) → Lim (rank‘(𝐴 × 𝐵)))
101	100	ex 450	. . . . . . 7 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (¬ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦 → Lim (rank‘(𝐴 × 𝐵))))
102	97, 101	orim12d 883	. . . . . 6 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → ((¬ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∨ ¬ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦) → (Lim (rank‘(𝐴 ∪ 𝐵)) ∨ Lim (rank‘(𝐴 × 𝐵)))))
103	79, 102	syl5bi 232	. . . . 5 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (¬ (∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦) → (Lim (rank‘(𝐴 ∪ 𝐵)) ∨ Lim (rank‘(𝐴 × 𝐵)))))
104	103	imp 445	. . . 4 ⊢ ((¬ (rank‘(𝐴 × 𝐵)) = ∅ ∧ ¬ (∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → (Lim (rank‘(𝐴 ∪ 𝐵)) ∨ Lim (rank‘(𝐴 × 𝐵))))
105		simpl 473	. . . . . . . 8 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ ¬ (rank‘(𝐴 × 𝐵)) = ∅) → Lim (rank‘(𝐴 ∪ 𝐵)))
106	34	necon3abii 2840	. . . . . . . . . 10 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
107	19, 20	rankxplim 8742	. . . . . . . . . 10 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
108	106, 107	sylan2br 493	. . . . . . . . 9 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ ¬ (rank‘(𝐴 × 𝐵)) = ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
109		limeq 5735	. . . . . . . . 9 ⊢ ((rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)) → (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴 ∪ 𝐵))))
110	108, 109	syl 17	. . . . . . . 8 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ ¬ (rank‘(𝐴 × 𝐵)) = ∅) → (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴 ∪ 𝐵))))
111	105, 110	mpbird 247	. . . . . . 7 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ ¬ (rank‘(𝐴 × 𝐵)) = ∅) → Lim (rank‘(𝐴 × 𝐵)))
112	111	expcom 451	. . . . . 6 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (Lim (rank‘(𝐴 ∪ 𝐵)) → Lim (rank‘(𝐴 × 𝐵))))
113		idd 24	. . . . . 6 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵))))
114	112, 113	jaod 395	. . . . 5 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → ((Lim (rank‘(𝐴 ∪ 𝐵)) ∨ Lim (rank‘(𝐴 × 𝐵))) → Lim (rank‘(𝐴 × 𝐵))))
115	114	adantr 481	. . . 4 ⊢ ((¬ (rank‘(𝐴 × 𝐵)) = ∅ ∧ ¬ (∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → ((Lim (rank‘(𝐴 ∪ 𝐵)) ∨ Lim (rank‘(𝐴 × 𝐵))) → Lim (rank‘(𝐴 × 𝐵))))
116	104, 115	mpd 15	. . 3 ⊢ ((¬ (rank‘(𝐴 × 𝐵)) = ∅ ∧ ¬ (∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → Lim (rank‘(𝐴 × 𝐵)))
117	8, 78, 116	syl2anc 693	. 2 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
118	1, 117	impbii 199	1 ⊢ (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim ∪ (rank‘(𝐴 × 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∨ w3o 1036 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 Vcvv 3200 ∪ cun 3572 ⊆ wss 3574 ∅c0 3915 ∪ cuni 4436 × cxp 5112 Ord word 5722 Oncon0 5723 Lim wlim 5724 suc csuc 5725 ‘cfv 5888 rankcrnk 8626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-reg 8497 ax-inf2 8538

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-r1 8627 df-rank 8628

This theorem is referenced by: rankxpsuc 8745

W3C validator